# Sum of Ideals is Ideal/Corollary

Jump to navigation
Jump to search

## Corollary to Sum of Ideals is Ideal

Let $J_1$ and $J_2$ be ideals of a ring $\struct {R, +, \circ}$.

Let $J = J_1 + J_2$ be an ideal of $R$ where $J_1 + J_2$ is as defined in subset product.

Then:

- $J_1 \subseteq J_1 + J_2$
- $J_2 \subseteq J_1 + J_2$

## Proof

From Sum of Ideals is Ideal we have that $j$ is an ideal of $R$.

Then:

- $0_R \in J_1 + J_2$

and so:

- $\forall x \in J_1: x + 0_R = x \in J_1 + J_2$

Similarly for $J_2$.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Theorem $40$